Schwartz very weak solutions for Schrödinger type equations with distributional coefficients
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912616218099712 |
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| author | Junior, Alexandre Arias Ascanelli, Alessia Cappiello, Marco Garetto, Claudia |
| author_facet | Junior, Alexandre Arias Ascanelli, Alessia Cappiello, Marco Garetto, Claudia |
| contents | This paper continues the analysis of Schrödinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schwartz very weak solutions for Schrödinger type equations with distributional coefficients Junior, Alexandre Arias Ascanelli, Alessia Cappiello, Marco Garetto, Claudia Analysis of PDEs 35J10, 35D99 This paper continues the analysis of Schrödinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data. |
| title | Schwartz very weak solutions for Schrödinger type equations with distributional coefficients |
| topic | Analysis of PDEs 35J10, 35D99 |
| url | https://arxiv.org/abs/2408.02642 |